`(2x^(4)+3x^(2)+1)/((x^(2)+1)^(4))=(A)/((x^(2)+1))+(B)/((x^(2)+1^(2)))+(C)/((x^(2)+1)^(3))+(D)/((x^(2)+1)^(4))` then Match the following.
`{:(" List - I"," List - II"),("1) A","(a) 2"),("2) B","(b) 1"),("3) C","(c) -1"),("4) D","(d) 0"), (,"(e) 1/2"):}`

Options

A. d, a, c, d
B. c, a, e, d
C. b, a, e, d
D. c, b, e, d

Answer

To solve the equation

\[ \frac{2x^4 + 3x^2 + 1}{(x^2 + 1)^4} = \frac{A}{(x^2 + 1)} + \frac{B}{(x^2 + 1)^2} + \frac{C}{(x^2 + 1)^3} + \frac{D}{(x^2 + 1)^4} \]

we will first express the left-hand side in terms of the right-hand side.

### Step 1: Set the equation
We start with the equation:

\[ 2x^4 + 3x^2 + 1 = A(x^2 + 1)^3 + B(x^2 + 1)^2 + C(x^2 + 1) + D \]

### Step 2: Expand the right-hand side
Now, we expand the right-hand side:

1. **Expand \(A(x^2 + 1)^3\)**:
\[ A(x^2 + 1)^3 = A(x^6 + 3x^4 + 3x^2 + 1) \]

2. **Expand \(B(x^2 + 1)^2\)**:
\[ B(x^2 + 1)^2 = B(x^4 + 2x^2 + 1) \]

3. **Expand \(C(x^2 + 1)\)**:
\[ C(x^2 + 1) = Cx^2 + C \]

4. **Combine all terms**:
\[ A(x^6 + 3x^4 + 3x^2 + 1) + B(x^4 + 2x^2 + 1) + Cx^2 + D \]

### Step 3: Combine like terms
Now we combine like terms:

- Coefficient of \(x^6\): \(A\)
- Coefficient of \(x^4\): \(3A + B\)
- Coefficient of \(x^2\): \(3A + 2B + C\)
- Constant term: \(A + B + C + D\)

### Step 4: Set up equations
Now we equate coefficients from both sides:

1. For \(x^6\):
\[ A = 0 \]

2. For \(x^4\):
\[ 3A + B = 2 \implies B = 2 \quad (\text{since } A = 0) \]

3. For \(x^2\):
\[ 3A + 2B + C = 3 \implies 2(2) + C = 3 \implies C = -1 \]

4. For the constant term:
\[ A + B + C + D = 1 \implies 0 + 2 - 1 + D = 1 \implies D = 0 \]

### Step 5: Collect results
We have found:
- \(A = 0\)
- \(B = 2\)
- \(C = -1\)
- \(D = 0\)

### Step 6: Match the results
Now we match the results with the given options:

- \(A = 0\) matches with (d) 0
- \(B = 2\) matches with (a) 2
- \(C = -1\) matches with (c) -1
- \(D = 0\) matches with (d) 0

### Final Matching
Thus, the matches are:
1. A → (d) 0
2. B → (a) 2
3. C → (c) -1
4. D → (d) 0